Mathematica Bohemica, Vol. 148, No. 1, pp. 11-34, 2023
Existence and stability results of nonlinear higher-order wave equation with a nonlinear source term and a delay term
Mama Abdelli, Abderrahmane Beniani, Nadia Mezouar, Ahmed Chahtou
Received August 21, 2020. Published online February 24, 2022.
Abstract: We consider the initial-boundary value problem for a nonlinear higher-order nonlinear hyperbolic equation in a bounded domain. The existence of global weak solutions for this problem is established by using the potential well theory combined with Faedo-Galarkin method. We also established the asymptotic behavior of global solutions as $t\rightarrow\infty$ by applying the Lyapunov method.
Keywords: nonlinear higher-order hyperbolic equation; nonlinear source term; global existence
References: [1] V. I. Arnold: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics 60. Springer, New York (1978). DOI 10.1007/978-1-4757-1693-1 | MR 0690288 | Zbl 0386.70001
[2] A. Benaissa, N. Louhibi: Global existence and energy decay of solutions to a nonlinear wave equation with a delay term. Georgian Math. J. 20 (2013), 1-24. DOI 10.1515/gmj-2013-0006 | MR 3037074 | Zbl 06152709
[3] P. Brenner, W. von Wahl: Global classical solutions of nonlinear wave equations. Math. Z. 176 (1981), 87-121. DOI 10.1007/BF01258907 | MR 0606174 | Zbl 0457.35059
[4] V. Komornik: Exact Controllability and Stabilization: The Multiplier Method. Research in Applied Mathematics 36. Wiley, Chichester (1994). MR 1359765 | Zbl 0937.93003
[5] J. L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Etudes mathematiques. Dunod, Paris (1969). (In French.) MR 0259693 | Zbl 0189.40603
[6] K. Liu: Locally distributed control and damping for the conservative systems. SIAM J. Control Optim. 35 (1997), 1574-1590. DOI 10.1137/S0363012995284928 | MR 1466917 | Zbl 0891.93016
[7] M. Nakao: Bounded, periodic and almost periodic classical solutions of some nonlinear wave equations with a dissipative term. J. Math. Soc. Japan 30 (1978), 375-394. DOI 10.2969/jmsj/03030375 | MR 0492914 | Zbl 0386.35004
[8] M. Nakao, H. Kuwahara: Decay estimates for some semilinear wave equations with degenerate dissipative terms. Funkc. Ekvacioj, Ser. Int. 30 (1987), 135-145. MR 0915268 | Zbl 0632.35046
[9] S. Nicaise, C. Pignotti: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45 (2006), 1561-1585. DOI 10.1137/060648891 | MR 2272156 | Zbl 1180.35095
[10] L. E. Payne, D. H. Sattinger: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math. 22 (1975), 273-303. DOI 10.1007/BF02761595 | MR 0402291 | Zbl 0317.35059
[11] H. Pecher: Die Existenz regulärer Lösungen für Cauchy- und Anfangs-Randwertprobleme nichtlinear Wellengleichungen. Math. Z. 140 (1974), 263-279. (In German.) DOI 10.1007/BF01214167 | MR 0364891 | Zbl 0287.35069
[12] B. Wang: Nonlinear scattering theory for a class of wave equations in $H^s$. J. Math. Anal. Appl. 296 (2004), 74-96. DOI 10.1016/j.jmaa.2004.03.050 | MR 2070494 | Zbl 1060.35099
[13] Y. Yanbing, M. S. Ahmed, Q. Lanlan, X. Runzhang: Global well-posedness of a class of fourth-order strongly damped nonlinear wave equation. Opusc. Math. 39 (2019), 297-313. DOI 10.7494/opmath.2019.39.2.297 | MR 3897819 | Zbl 1437.35454
[14] Y. Ye: Existence and asymptotic behavior of global solutions for a class of nonlinear higher-order wave equation. J. Inequal. Appl. 2010 (2010), Article ID 394859, 14 pages. DOI 10.1155/2010/394859 | MR 2600191 | Zbl 1190.35161
[15] E. Zuazua: Exponential decay for the semilinear wave equation with locally distributed damping. Commun. Partial Differ. Equations 15 (1990), 205-235. DOI 10.1080/03605309908820684 | MR 1032629 | Zbl 0716.35010
Affiliations: Mama Abdelli, Laboratory ACEDP, Djillali Liabes University, P.O. Box 89, Sidi Bel Abbes 22000, Algeria, e-mail: abdelli.mama@gmail.com; Abderrahmane Beniani, EDPs Analysis and Control Laboratory, Department of Mathematics, BP 284, University Ain Témouchent BELHADJ Bouchaib, Ain Témouchent 46000, Algeria, e-mail: a.beniani@yahoo.fr; Nadia Mezouar, Ahmed Chahtou, Laboratory ACEDP, Djillali Liabes University, P.O. Box 89, Sidi Bel Abbes 22000, Algeria, e-mail: nadiamezouar1980@gmail.com, hmidamath@gmail.com
